On two notions of a Gerbe over a stack

Let $mathcal{G}$ be a Lie groupoid. The category $Bmathcal{G}$ of principal $mathcal{G}$-bundles defines a differentiable stack. On the other hand, given a differentiable stack $mathcal{D}$, there exists a Lie groupoid $mathcal{H}$ such that $Bmathcal{H}$ is isomorphic to $mathcal{D}$. Define a gerbe over a stack as a morphism of stacks $Fcolon mathcal{D}rightarrow mathcal{C}$, such that $F$ and the diagonal map $Delta_Fcolon mathcal{D}rightarrow mathcal{D}times_{mathcal{C}}mathcal{D}$ are epimorphisms. This paper explores the relationship between a gerbe defined above and a Morita equivalence class of a Lie groupoid extension.